A Minkowski-type theorem on distances to cusps: the general case

In a previous paper, we studied the connection between points in $\mathbb{H}^n$ and $2$-dimensional rigid adelic spaces on a totally real number field $K$ with class number $h_K = 1$. This last assumption was needed to link heights and distances to cusps. In this paper, we remove this hypothesis to obtain, without restriction on $K$ totally real, an analogue of Minkowski's second theorem on the Roy--Thunder minima of a $2$-dimensional rigid adelic space in the framework of distances between a point $\tau \in \mathbb{H}^n$ and its two closest cusps.